#7560629

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,27)(17,21)(18,22)(20,29)(23,35)(24,38)(25,34)(26,33)(28,32)(30,41)(31,37)(36,42)(39,43)(40,44);(2,3)(6,9)(10,12)(13,15,17,25)(14,19,21,34)(16,28,29,23)(18,24,31,26)(20,35,27,32)(22,38,37,33)(30,39,42,40)(36,44,41,43);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,29)(18,30)(19,32)(21,27)(22,36)(24,40)(25,28)(26,39)(31,42)(33,44)(34,35)(37,41)(38,43);(1,3,4,2)(14,21)(15,24)(16,29)(18,31)(19,33)(23,39)(25,26)(28,40)(32,43)(34,38)(35,44)(36,41);(6,9)(10,12)(14,21)(16,29)(19,34)(22,37)(23,28)(30,42)(33,38)(39,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467825);